Download presentation on round bodies. Presentation on the topic: Round geometric bodies. Historical note about the cylinder

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Historical background about the cone

CYLINDER.. The word "cylinder" comes from the Greek kylindros, which means "roller", "skating rink". CONE. The Latin word conus is borrowed from the Greek language (konos - plug, sleeve, pine cone). In the XI book of the "Beginnings" the following definition is given: if a right-angled triangle rotating around one of its legs returns to the same position from which it began to move, then the described figure will be a cone. Euclid considers only

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Sphere A sphere is a surface consisting of all points in space located at a given distance from a given point. A radius-segment connecting the center to any point of the sphere A diameter-segment connecting two points of the sphere and passing through its center. A chord is a segment that connects any two points on a sphere.

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Sphere area

For the area of ​​the sphere, we take the limit of the sequence of areas of the surfaces of polyhedra circumscribed around the sphere as the largest size of each face tends to zero. S=4πR^2

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Tangent plane to sphere

A tangent plane to a sphere is a plane that has only one common point with the sphere. The point of contact is their common point. Theorem: The radius of the sphere, drawn to the point of contact between the sphere and the plane, is perpendicular to the tangent plane. Theorem: If the radius of a sphere is perpendicular to a plane passing through its end lying on the sphere, then this plane is tangent to the sphere

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Historical background about the sphere

However, both the words "ball" and "sphere" come from the same Greek word "sfire" - ball. At the same time, the word "ball" was formed from the transition of consonants sph into sh. In ancient times, the sphere was held in high esteem. Astronomical observations of the firmament invariably evoke the image of a sphere. The Pythagoreans taught about the existence of ten spheres of the Universe, along which celestial bodies allegedly move. They argued that the distances of these bodies from each other are proportional to the intervals of the musical scale. In this they saw the elements of world harmony. Pythagorean "music of the spheres" was contained in such semi-mystical reasoning. Aristotle believed that the spherical shape, as the most perfect, is characteristic of the Moon, the Sun, the Earth and all world bodies. Developing the views of Eudoxus, he believed that the Earth is surrounded by a series of concentric spheres. The sphere has always been widely used in various areas science and technology. In Book XI of the Elements, Euclid defines a sphere as a figure described by a semicircle rotating about a fixed diameter.

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Vodovzvodnaya Tower The Vodovzvodnaya Tower was built in 1488. The former name of the tower - Sviblova - is associated with the courtyard of the boyar Sviblova located nearby. In 1633, a water pump was installed in the tower to pump water into a reservoir located on the top of the tower. Through the pipes, water dispersed throughout the Kremlin. In 1805-1806, the tower was dismantled and rebuilt according to the project of the architect I.V. Egotov. In 1812, the tower was blown up by the French, and in 1819 it was restored under the leadership of O.I. Bove. The height of the tower to the star is 57.7 meters, with the star - 61.25 meters. The tower is a cylinder. The tower is round in cross section.

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Krivoarbatsky pereulok, building 10. Two huge white cylinders leaning against each other. Along the perimeter - sixty small diamond-shaped windows, creating the image of a beehive. On the facade there is a giant window several meters high. Above the window there is an inscription: "Konstantin Melnikov. Architect". The most famous (even iconic) building of the 1920s in Moscow. Konstantin Stepanovich Melnikov was born in Moscow in 1890 into a family of a construction worker, who came from a peasant family. Chaplin helped him enter in 1905. B Moscow School of Painting, Sculpture and Architecture, and then after graduating from Melnikov in 1913. painting department advised to continue his studies at the Architectural Department, which Konstantin Stepanovich graduated in 1917. In the senior years of the College and in the first years after graduation, Melnikov worked in the spirit of neoclassicism. However, already in the early 1920s, Konstantin Stepanovich broke sharply with various kinds of traditionalist stylizations. The very fact of the wide realization of his works makes us take a different attitude towards those of his works that remained in the projects and which in the 1920s, in the sharp controversy of that period, were often declared "fantastic". In Melnikov's projects, the degree of uninhibitedness of the master's creative imagination in matters of shaping is striking. It can be said with full confidence that in the XX century. there was no other architect who would create so many fundamentally new projects and such a level of novelty that their originality not only severely separated them from the works of other masters, but also differed just as much from the works of their author himself.

The date: 23.12.2017

Teacher: Kuksenko Natalia Nikolaevna

Subject: maths

Class: 6

Topic: Round bodies

Formed UUD: the ability to plan ways to achieve the intended goals; the ability to adequately assess the degree of objective and subjective difficulty in completing a learning task

Target: to introduce students to geometric bodies: a ball, a cone, a cylinder - and their elements.

Tasks:

be able to operate with concepts: ball, cone, cylinder, base, height, vertex, sphere, center, radius, diameter, circular sector, section when performing various tasks; be able to recognize the studied geometric shapes; be able to give examples of objects that have the shape of the studied bodies of revolution; be able to talk about a ball, cone, cylinder according to plan.

During the classes:

Update

Oral survey.

1. The radii of the circles are 3 cm and 5 cm. What is their relative position if the distance between the centers is

a) 8 cm?; b) 10 cm; c) 6 cm; d) 0

2. Name equal elements in triangles.

a)

b)

2. Problematization (learning task)

Correctly read the statement, written without spaces: Mathematics is the queen of all sciences. Her beloved is truth, her people are simplicity and clarity. The palace of this mistress is surrounded by thorny thickets, and in order to achieve it, everyone has to wade through the thicket. A casual traveler will not find the water of the palace for nothing attractive.

goal setting

In this lesson, you will be introduced to three new geometric shapes. To better understand new material, be attentive, active and quick-witted. The topic of the lesson is encrypted with the help of puzzles. Solve them and you will find out what geometric shapes we will study today.

So, the topic of the lesson is "Round Bodies"

- Write the topic of the lesson in your notebook.

What is the purpose of our lesson?

4. Main body

1) Remember which figure was encrypted in the rebus in the form of a hat?

What other objects are cylindrical?

It turns out that the word "cylinder" comes from the Greek word "kyulindros", meaning "roller", "skating rink".

At the turn of the 18th - 19th centuries, men in many countries wore hard hats with small fields, which were called cylinders because of their great resemblance to the geometric figure of a cylinder.

Let's take a closer look at the cylinder (demonstration of the model) and see that the cylinder consists of two bases located in parallel planes and a side surface.

The cylinder is obtained by rotating a rectangle around one of its sides.

What are the bases of a cylinder?

What can you say about the size of these circles?

What is the side surface of a cylinder?

Look at the cylinder bore. What is the lateral surface of a cylinder?

The cylinder has parameters - this is the height and radius.

Let's try to formulate the definition of the height and radius of the cylinder.

So, height is a segment connecting the centers of the bases, perpendicular to each of them; cylinder radius - the radius of the circle that is the base of the cylinder.

Practical task.

Fold the side surface of the cylinder from a rectangular sheet. What is its height?

Imagine a situation where we need to cut a cylinder.

How can this be done and what happens in the cross section of the cylinder?

2) - And now we turn to the consideration of the cone.

The word "cone" comes from the Greek word "konos", meaning a pine cone (showing a cone). Indeed, there are some similarities.

What objects are shaped like a cone?

The cone consists of a base and a side surface.

A cone is obtained by rotating a right triangle around its side with a right angle.

What is the base of the cone?

What is the side surface?

What the side surface is like, we will see by turning the paper cone onto a plane. The lateral surface of the cone unfolds into a circular sector - a part of a circle bounded by two radii.

A cone has a vertex, height, base radius

Let's formulate a definition.

So, the height is a perpendicular drawn from the top of the cone to the center of the base.

If we cut off the vertex and the upper part of the cone (I show it on the model), then we will get the so-called truncated cone.

- Think and say, what objects have the shape of a cone or a truncated cone?

How is it possible to cut the cone and what happens in its cross section?

It turns out that the sections of the cone can have the shapes of other geometric shapes, the names of which we don’t even know yet, we will study them in high school, and therefore we won’t talk about them yet

3) -Let's move on to the study of the ball.

Give examples of surrounding objects that have the shape of a ball.

What do you think a ball and a circle and a circle have in common?

The ball is obtained by rotating a semicircle around the diameter.

The surface of the sphere is calledsphere.The word "sphere" comes from the Greek word "sfire", which is translated into Russian as "ball". Do not confuse the concepts of "ball" and "sphere". A sphere is, one might say, a shell or boundary of a sphere.

A ball, a globe are spheres, but a watermelon, an orange, the Sun, the Moon, the Earth and other planets have the shape of a slightly flattened ball (the picture shows).

Try to call the sections of the ball planes.

Which section will be the largest?

So, we got acquainted with three spatial figures, otherwise they are called geometric bodies. In 5th grade, you met polyhedrons. Let's remember their names.

Why are they called polyhedrons?

What would you call the new geometric bodies?

Indeed, all geometric bodies are divided into two groups: polyhedra and bodies of revolution.

Working with the textbook

7. Evaluation

- Summarize knowledge by completing a test in a notebook.

Task number 1. What form of objects is the tower made of? Name from top to bottom.

(Cone, cube, cylinder)

Task number 2. The figure shows various geometric bodies. Which of them are polyhedra?

Second (pyramid), third (tilted prism)

Task number 3. In the figure, the first line shows the front view of the figure, and the second line shows the top view of the figure. What is this figure?

1. Cone. 2.Cylinder. 3. Quadrangular pyramid. 4. Rectangular parallelepiped. 5. Triangular pyramid. 6. Ball.

Task number 4. There are three cones of different colors on the round table - red, blue and green. Children sit around the table: Masha, Vanya, Dasha, Kolya, Raya and Petya. Which of the children sees such a picture, as shown in the picture under the letter a); b); in)?

a B C)

(Petya) (Vanya) (Masha)

Task number 5. The figure shows some geometric bodies. Perhaps the point of view is not very familiar. What bodies, when viewed from the appropriate side, might look like the one in the figure? Which of the drawings can correspond to the same body?

1. Cube or box. 2. Pyramid or cone. 3. Cone, cylinder or ball. 4. Parallelepiped. Figures 2 and 3 may correspond to a cone, and figures 1 and 4 to a parallelepiped.

8. Reflection

If you think you understand the topic of the lesson, then draw a green circle.

If you think that you have not mastered the material enough, then draw a blue circle.

If you think you didn't understand the topic of the lesson, then draw a red circle.

9. Perspective (homework) № 446, 448